We consider connections between similar sublattices and coincidence site lattices (CSLs), and more generally between similar submodules and coincidence site modules of general (free) $mathbb{Z}$-modules in $mathbb{R}^d$. In particular, we generalis
e results obtained by S. Glied and M. Baake [1,2] on similarity and coincidence isometries of lattices and certain lattice-like modules called $mathcal{S}$-modules. An important result is that the factor group $mathrm{OS}(M)/mathrm{OC}(M)$ is Abelian for arbitrary $mathbb{Z}$-modules $M$, where $mathrm{OS}(M)$ and $mathrm{OC}(M)$ are the groups of similar and coincidence isometries, respectively. In addition, we derive various relations between the indices of CSLs and their corresponding similar sublattices. [1] S. Glied, M. Baake, Similarity versus coincidence rotations of lattices, Z. Krist. 223, 770--772 (2008). DOI: 10.1524/zkri.2008.1054 [2] S. Glied, Similarity and coincidence isometries for modules, Can. Math. Bull. 55, 98--107 (2011). DOI: 10.4153/CMB-2011-076-x
Recently, the group of coincidence isometries of the root lattice $A_4$ has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since diffe
rent coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions $d leq 3$, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but nevertheless give rise to the same CSL. We indicate how the classification of CSLs can be obtained by making use of the icosian ring and provide explicit criteria for two isometries to generate the same CSL. Moreover, we determine the number of CSLs of a given index and encapsulate the result in a Dirichlet series generating function.
Coincidence Site Lattices (CSLs) are a well established tool in the theory of grain boundaries. For several lattices up to dimension $d=4$, the CSLs are known explicitly as well as their indices and multiplicity functions. Many of them share a partic
ular property: their multiplicity functions are multiplicative. We show how multiplicativity is connected to certain decompositions of CSLs and the corresponding coincidence rotations and present some criteria for multiplicativity. In general, however, multiplicativity is violated, while supermultiplicativity still holds.
